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In this paper we study the physical and statistical properties of theperiodic Lorentz gas with finite horizon driven to a non-equilibrium steadystate by the combination of non-conservative external forces and deterministicthermostats. A version of this model was introduced by Chernov, Eyink,Lebowitz, and Sinai and subsequently generalized by Chernov and the thirdauthor. Non-equilibrium steady states for these models are SRB measures andthey are characterized by the positivity of the steady state entropy productionrate. Our main result is to establish that the entropy production, in thiscontext equal to the phase space contraction, satisfies the Gallavotti-Cohenfluctuation relation. The main tool needed in the proof is the family ofanisotropic Banach spaces introduced by the first and third authors to studythe ergodic and statistical properties of billiards using transfer operatortechniques.

Fluctuation of the entropy production for the Lorentz gas under small external forces